Export Model With Material Colors to MeshLab
For importing the mesh with colors into MeshLab:

In Blender, select File > Export > Wavefront (.obj), and select a location to save the .obj to.



When the file browser opens as part of the export process, look at the panel to the left, and make sure "Write materials" is checked:



Next, open MeshLab, and choose File > Import Mesh... and select the .obj you exported from Blender. This will import the mesh, but the mesh will be grey: 



To enable the mesh colors you exported from Blender, in MeshLab you need to navigate to Render > Color and click "Per Face". Your mesh will now be displayed with the colors you specified in Blender:



For the second part of your question, you should be able to select the black wireframe by right-clicking on it, and then delete it by pressing X. Does that achieve what you need? If that doesn't work, or if you encounter problems, could you upload your .blend file please?